One of the things I never understood, but was too afraid to ask is this: how should I think of things like $\frac{\text{kg}}{\text{s}^2}$. What exactly is a square second? Square foot makes sense to me because I can see it, but square second? I've always assumed it's just one of those things you're supposed to deal with abstractly and not try to get an intuition for, the same way you deal with 4-dimensional space. Should I just keep doing that or is there some bit of wisdom I've been deprived of all these years?

3 Answers
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Every now and then I run into this question from students. I approach it as follows:

In computing how much paint you will need to cover a wall, it's natural to think in terms of square meters $m^2$. Students seem to accept this intuitively. The problem arises when having to think about things like time squared.

(1) When you use units of square meters you must recognize that this is already a derived unit. You don't use a square meter to measure square meters. You use a meter stick, and then you use some math. So you already lost your virginity in square meters, square seconds is just another step on the path.

(2) When you go down to buy a car (in the US), one of the attributes they will sell you the car on, in terms of measuring its acceleration, is how many seconds it takes to reach a speed, typically 60 mph. So a car might take 10 seconds to reach 60 miles per hour. Assuming the acceleration is constant, this is 6 miles per hour each second. We write this as
$$\frac{\textrm{60 miles per hour}}{\textrm{10 seconds}} =
\frac{\textrm{6 miles per hour}}{\textrm{second}}.$$
And this can be rewritten as:
$$\frac{\textrm{6 miles /hour}}{\textrm{second}}
= \frac{\textrm{6 miles /hour}}{\textrm{second/1}}$$
$$= \frac{\textrm{6 miles}}{\textrm{hour}} \frac{\textrm{1}}{\textrm{second}}$$
$$= \frac{\textrm{6 miles}}{\textrm{hour-second}}$$
and so we are left with the unit "hour-second" which indeed is time squared, and is easily seen to be equal to minute$^2$.

In short, we must always remember that the units we use are there only to allow us to make calculations. We defined them, we use them, nature abides by them, but She does not contain them.

For many compound units, you shouldn't try to "visualize" the meaning of the unit, but you should think of it as reminding you about relationships between that quantity and others. Why are the units of Newton's constant $G$ ${\rm N\ m^2/kg^2}$? It's because $G$'s "purpose in life" is to be multiplied by a couple of masses and divided by a squared distance, leaving you with a force.

By the way, this sometimes means that, at least when you're new to a quantity, it's often nice not to reduce its units to the simplest form. Lubos's example is probably the best one here: The meaning of ${\rm m/s^2}$ is obscure to some beginning physics students, whereas $\rm (m/s)/s$ or
$${\rm m/s}\over {\rm s}$$
is a clearer reminder of the meaning. (If you write it in the first form, please use the parentheses. Older books used to write the potentially ambiguous m/s/s.) Similarly, I wrote the units of $G$ in the form I did because that's the way it's easy to remember. It's equivalent to $\rm m^3/(kg\ s^2)$, but the "meaning" of this is harder to see at a glance.

There is no reason why you should be "imagining" a squared second. Most quantities in physics don't have any canonical "geometric" visualization and there is no reason why they should have. What matters is that you should be able to calculate with it.

For example, the gravitational acceleration on Earth is $9.81\,\,{\rm m/s}^2$. This simply means that the acceleration is
$$ g = \frac{9.81 \,\,{\rm m/s}}{{\rm s}}$$
Every second, the velocity increases by $9.81 \,\,{\rm m/s}$ in the downward direction. The acceleration is 9.81 meters per second per second. If you divide the unit ${\rm m/s}$ by another ${\rm s}$, you get ${\rm m/s/s}$ which is the same thing as ${\rm m/s}^2$.

A squared second would still be very simple to imagine: just imagine a square in a fictitious spacetime with two time coordinates whose side is one second. There's no problem with the fact that this spacetime is not real: you are just trying to imagine something that shouldn't be imagined, so it's not surprising that the imaginations are unphysical.

There exist much more "bizarre" units for seemingly simple quantities. For example, the unit of electric charge in the CGSE system is one statcoulomb

which is just a different way of saying
$$ 1 {\rm g}^{1/2} {\rm cm}^{3/2} {\rm s}^{-1} $$
which contains fractional powers. You can't imagine any shapes whose "volumes" are fractional powers of the sides. Still, there is no difficulty with calculating with these units. There are lots of formulae in physics which are "nonlinear" - in which one quantity has to be inverted, squared, cubed, or exponentiated to another (possibly fractional) power - to obtain another quantity. The units must also be exponentiated accordingly.